This code does what you need:
Dim FileLink As String = "yourserver/file.txt" 'Link to your text file
Dim Destination As String = "C:\" 'Link to the download directory
Dim Client As WebClient = New WebClient() 'New web client
Dim Reader As StreamReader = New StreamReader(Client.OpenRead(FileLink)) 'A stream reader to get your file contents
Dim A As String = Reader.ReadToEnd 'Outputing the file contents to A
Dim Phase As Integer = 0 'A phase which determens which line you're at
Dim Links(500) As String 'A string array to hold 500 links, you can edit the number
For I As Integer = 0 To A.Length 'Looping through A
If A(I) = Environment.NewLine Then 'If the current character is a newline, it'll start downloading the from the first link and increase the Phase
My.Computer.Network.DownloadFile(Links(Phase), Destination, "", "", False, 9800, True) 'Link, destination, id, password, show UI or not, timeout, overwrite or not
Phase += 1
Else 'If it's not a newline then the link is still incomplete, it adds the character to the link currently being written
Links(Phase) += A(I)
End If
Next